-
Under consideration for publication in Euro. Jnl of Applied
Mathematics 1
Two-dimensional Stokes flow with suction andsmall surface
tension
L.J. CUMMINGS1 and S.D. HOWISON2
1 Labo. de Phys. Stat. de l’E.N.S., 24 rue Lhomond, 75231 Paris
Cedex 05, France2 Mathematical Institute, 24-29 St Giles’, Oxford
OX1 3LB, England.
(Received 10 April 2006)
In this article the complex variable theory of two-dimensional
Stokes flow as developed by Richard-
son [23], and modified by Howison & Richardson [17], is
reviewed. The analysis of [17] is extended
to a new solution, which uses a cubic polynomial conformal
mapping (with real coefficients) from
the unit disc onto the fluid domain. An apparent “stability
paradox” (where two equivalent flow
geometries are found, one of which is ‘stable’ and the other
unstable) is resolved by allowing the
coefficients to take complex values.
1 Introduction
The problem of two-dimensional slow viscous flow (Stokes flow)
with a free boundary under surfacetension has been the subject of
much research in the last few years.1 It has long been known
thatmany exact steady solutions exist, for both viscous “blobs” and
for bubbles in unbounded fluiddomains (for instance, Richardson
[21, 22], Garabedian [9], Buckmaster [2]). The recent intereststems
from the remarkable discovery that many exact time-dependent
solutions to the problemcan be found in closed form, using ideas
from complex variable theory [11–13,23].
More recently still, Howison & Richardson [17] considered
time-evolving problems incorporatinga driving mechanism (a single
point sink) at some finite point within the (bounded) fluid
domain,and Tanveer & Vasconcelos [26] solved for similar
problems but with bubbles evolving in timeunder the action of some
prescribed flow at infinity. Stokes flows with singularities have
beenconsidered before, but to the authors’ knowledge, all such
treatments have been for the steadyproblem; see for instance
[1,19], where the flow is sustained by a vortex dipole singularity.
Whenwe allow such singularities in the unsteady problem, the
possibility is raised of considering the(much easier) zero surface
tension (ZST) problem.
This problem admits very many simple exact solutions, and in
physical situations where therelevant dimensionless surface tension
parameter T (which is a measure of the effects of surfacetension
versus those of the driving mechanism) is small, these may provide
a good approximationto the actual solution. However, the ZST
problem has the drawback that its solutions can undergofinite time
blow-up, with curvature singularities (often, but not always,
cusps) appearing in thefree boundary—this is the case for the
point-sink driven problems considered in [17] (cf. theZST Hele-Shaw
problem with suction, also discussed in [17]). The approximate ZST
boundarycondition becomes invalid as such a curvature singularity
develops. Nevertheless, for small valuesof the surface tension
parameter, the evolution of the ZST solution prior to blow-up gives
a goodapproximation to the solution with positive surface tension.
In such problems, it seems reasonableto consider the evolution in
two parts, assuming “ZST theory” until we near blow-up time,
andonly then introducing surface tension effects. Experimental
evidence, showing that very highly-
1 Stokes flow is of practical importance in many industrial
processes, some of which are listed in [26],and discussed in
references therein.
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2 L.J. Cummings and S.D. Howison
curved interfaces can be sustained in driven flows, supports
this idea [2, 19]. In [17] the authorstake this idea to its logical
conclusion, considering the limit T = 0+. We emphasise that
thedifficulties encountered (due to the singular limit T → 0+) when
attempting to regularise theZST Hele-Shaw problem by introducing
small positive surface tension, do not arise for Stokesflow. We
also point out that for Stokes flow, before singularity formation
the ZST problem isidentical to the zero-surface-tension limit of
the problem with T > 0. This question remainsunresolved at a
rigorous level for the Hele-Shaw problem, although [25] suggests
that in this casethe limiting regularised problem can differ from
the unregularised problem at an O(1) time beforethat of singularity
formation.
This paper explores this concept further, applying it to analyse
a more complicated example.Our initial “simplest case” analysis, in
§5, throws up an apparent paradox, the resolution of which(in §5.2)
reveals a surprisingly rich solution structure. The results we
obtain are of interest bothin the general study of (experimentally
observable) cusp formation in viscous flow [1,2,19,20], andalso in
the wider mathematical context of conformal mapping techniques
applied to free boundaryproblems.
2 Complex variable theory
We begin by outlining the theory as developed in [17, 23]. The
notation we employ is largelythe same as those papers. It is
convenient to work with the streamfunction ψ(x, y, t), which
intwo-dimensional slow viscous flow satisfies the biharmonic
equation
∇4ψ = 0.The fluid is assumed to occupy some simply-connected
domain Ω(t) in R2, which we can consideras a subset of C if we
write z = x+ iy.
As well as the kinematic boundary condition (equating the
velocity of the fluid at the freeboundary, in the direction of the
outward normal, to the velocity of the free boundary itself inthat
direction) there are two stress boundary conditions,
σijnj = −Tκni i = 1, 2, on ∂Ω(t). (2.1)Here, σij denotes the
usual Newtonian stress tensor, n = (ni) denotes the outward normal
to∂Ω(t), T is a surface tension coefficient, and κ is the curvature
of the free boundary.
Using the Goursat representation of biharmonic functions ψ,
together with its biharmonic con-jugate the Airy stress function A,
may be expressed in the form
A+ iψ = W(z, z̄, t) = −(z̄φ(z, t) + χ(z, t)),for functions φ(z,
t), χ(z, t) analytic on the flow domain except at driving
singularities. In thispaper we consider only flows driven by a
point sink (or source) of strength Q at the origin, whereφ and χ
have the local behaviour
φ(z, t) = O(1), χ′(z, t) =Q
2πz+O(1), as z → 0; (2.2)
if Q < 0 we have a point source. (We use ′ for ∂/∂z or ∂/∂ζ
without ambiguity.) All physicalquantities of interest can be
expressed in terms of the functions φ and χ; for instance, the
velocityfield (u, v) and the pressure field p are given by
u+ iv = φ(z, t)− zφ′(z, t)− χ′(z, t), p = −4µ
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Two-dimensional Stokes flow with suction and small surface
tension 3
We now introduce a time-dependent conformal map f(ζ, t) from the
unit disc in ζ-space onto thefluid domain. This is always possible
by the Riemann mapping theorem, and the map is uniquelydetermined
if we insist f(0, t) = 0, f ′(0, t) > 0 for all time. We define
the functions2 Φ(ζ, t) andX (ζ, t) on the unit disc by
Φ(ζ, t) = φ(f(ζ, t), t), X (ζ, t) = χ(f(ζ, t), t).The boundary
conditions (2.1) and the kinematic boundary condition may now be
formulated inthe ζ-plane, as identities holding on |ζ| = 1. We
refer to Richardson [23] for the details; briefly,when the (purely
mathematical) condition Φ(0, t) = 0 is assumed, these identities
are easilyanalytically continued, giving functional identities
which hold globally in the ζ-plane (equations(2.18) and (2.19) of
[23]).3 These equations are most conveniently expressed as follows
(see forexample [6, 7]):
∂
∂t
(f ′(ζ, t)f̄(1/ζ, t)
)+ 2X ′(ζ, t) = T
2µ∂
∂ζ
(ζf ′(ζ, t)f̄(1/ζ, t)G+(ζ, t)
), (2.3)
and
2Φ(ζ, t)− ∂f∂t
(ζ, t) +T
2µG+(ζ, t)ζf ′(ζ, t) = 0, (2.4)
where the function G+(ζ, t) (analytic on |ζ| ≤ 1) is defined in
terms of the conformal map via
G+(ζ, t) =1
2πi
∮
|τ |=1
1|f ′(τ, t)|
τ + ζτ − ζ
dτ
τ. (2.5)
Hence the solution procedure entails a search for conformal maps
f(ζ, t) satisfying (2.3) and (2.4),together with appropriate
behaviour at the driving singularities. For the case of a single
pointsink at the origin this behaviour is
X (ζ, t) = Q2π
log ζ +O(1) as ζ → 0 (2.6)
(Φ(0, t) = 0 by assumption). Once one has postulated a suitable
form for f(ζ, t), one can find asolution to the problem by matching
the singular behaviour on either side of the equation (2.3).For our
solution we shall use a slightly different approach however, based
on results outlined in§3, which is particularly appropriate for
polynomial mapping functions.
A crucial requirement of the theory (on both physical and
mathematical grounds) is that themapping function f(ζ, t) be
univalent on the unit disc; solutions break down when this
conditionis violated. See [8] for a detailed discussion; loosely
speaking, f is required to be analytic and one-to-one, with
nonvanishing derivative (conformal). If we assume an initially
univalent map f(ζ, 0),then any such nonunivalency must approach |ζ|
= 1 from without, hence nonunivalencies are firstmanifested on the
boundary of the domain, in both ζ-space and z-space (physical
space). Forinstance, if the mapping becomes double-valued at some
point on the unit circle, this correspondsto the fluid domain
beginning to overlap itself, at which point the model must be
modified toaccount for the change in topology. A zero in the
derivative of f on the unit circle corresponds toa cusp (or,
exceptionally, a corner) in the free boundary, which we might
expect to be smoothedoff by the action of positive surface tension.
We refer forward to Figure 2 for examples of self-overlapping and
cusped flow domains.
The conformal maps we shall use contain various time-dependent
parameters, not all values ofwhich give rise to a univalent map on
|ζ| ≤ 1. In general there is a subset of the parameter space on
2 This notation differs from that of Richardson [23] since he
uses w(ζ, t) rather than f(ζ, t) for theconformal map, and takes
X(ζ, t) = χ′(w(ζ, t), t), which may lead to confusion.
3 Imposition of this condition can lead to solutions which do
not conserve overall momentum (orequivalently, which have moving
singularities within the flow). We refer to [7] or [17] for further
discussionof this point.
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4 L.J. Cummings and S.D. Howison
which the map is univalent on the unit disc, and to which we
must restrict attention when solvingproblems. The idea of such a
“univalency domain” (which we denote by V ) in parameter spaceis an
important concept in this paper, since one way to represent the
solution is as a trajectoryfollowed within V as the parameters
evolve in time. A trajectory reaching the boundary of V isthen
associated with blow-up of the solution, by one of the means listed
above. (See [18] for theapplication of similar ideas in the context
of the Hele-Shaw free boundary problem.)
3 Stokes flow “moments”
Consider the quantities Ck(t) defined (for integers k ≥ 0)
by
Ck(t) =∫ ∫
Ω(t)
ζ(z, t)k dxdy =12i
∫
∂Ω(t)
ζ(z, t)k z̄dz =12i
∫
|ζ|=1ζkf ′(ζ, t)f̄(1/ζ, t) dζ. (3.1)
It follows from equation (2.3) (see [6] for the details; also
[5] where similar ideas are applied) thatwith only the singularity
(2.6) at the origin,
dC0(t)dt
= −Q, (3.2)
dCk(t)dt
= −kT2µ
∞∑r=0
G(r)+ (0, t)r!
Ck+r(t), k ≥ 1. (3.3)
The quantity C0(t) is clearly the area of the fluid domain, so
the k = 0 equation just representsconservation of mass. Note that
the ZST case is particularly simple, leading to an infinite set
ofconserved quantities.
For a general polynomial mapping function,
f(ζ, t) =N∑1
ar(t)ζr, (3.4)
the Ck(t) are readily evaluated from (3.1) as
Ck = πN−k∑n=1
nanān+k 0 ≤ k ≤ N − 1, (3.5)
all other Ck being identically zero (note that this shows that
if f(ζ, 0) is a polynomial of degree nthen it must remain so for t
> 0). Hence we obtain a system of N equations, (3.2) and (3.3),
forthe nonzero Ck, which are equivalent to a system of equations
for the N coefficients ak(t). In theZST case these are algebraic,
but the general NZST equations are a system of nonlinear
ordinarydifferential equations.
4 The limit of zero surface tension
It is clear from the above that the ZST problem is very much
simpler than the NZST one.Nonetheless, several time-dependent
problems have been solved exactly using conformal mappingideas
similar to those outlined here, including:
(1) The coalescence under surface tension of two (equal or
unequal) circular cylinders of fluid[11,23];
(2) The coalescence under surface tension of a cylinder and a
half-space of fluid [13];
(3) A limaçon evolving under the action of surface tension only
[23];
(4) The evolution of domains described by polynomial maps of the
form
f(ζ, t) = a(t)(ζ − b(t)ζn/n)
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Two-dimensional Stokes flow with suction and small surface
tension 5
for any integer n ≥ 2, under the action of both surface tension
and a point sink at theorigin [17] ((3) above is a special case of
this solution family, with n = 2 and no suction);
(5) The evolution of bubbles in a shear flow with surface
tension effects included [26];
(6) The evolution of expanding/contracting bubbles in otherwise
quiescent flow with surfacetension effects included [26].
The work that we are concerned with here is that of (4) above,
due to Howison & Richardson( [17]; henceforth HR’95), since
they include the effects of both surface tension and a
drivingmechanism; we thus give a short review of their work. They
consider fluid domains Ω(t) driven bya single point sink of
strength Q > 0 at the origin, which are described by the family
of mappingfunctions
z = f(ζ, t) = a(t)(ζ − b(t)
nζn
), |ζ| ≤ 1, (4.1)
for a(t), b(t) real and positive functions of time, and integers
n ≥ 2. The maps (4.1) are univalent onthe unit disc only if b(t)
< 1, with (n−1) inward-pointing 3/2-power cusps forming
simultaneouslyon ∂Ω(t) if b(t) = 1.
We use the results of §3. By (3.5) the only nonzero moments are
C0 and Cn−1. The evolutionof these quantities is given by (3.2) and
(3.3), hence we must calculate G+(0) using (2.5). This
isstraightforward, leading to the evolution equations
dS
dt=
d
dt
[πa2
(1 +
b2
n
)]= −Q, (4.2)
d
dt(a2b) = −(n− 1) T
πµabK(b), (4.3)
where S(t) denotes the cross-sectional area of Ω(t), and K( · )
denotes the complete elliptic integralof the first kind (see (B 4)
in Appendix B, with φ = π/2).
Howison & Richardson considered an (a, b) phase plane within
which the univalency domainis 0 ≤ b < 1, a ≥ 0 (recall the
comments at the end of §2); a solution trajectory (a(t),
b(t))reaching the univalency boundary b = 1 is associated with
formation of 3/2-power cusps. Solutionbreakdown before extraction
is inevitable when T = 0, with b(t∗) = 1, a(t∗) > 0, for
somepositive “blow-up” time t∗. However solving (4.2) and (4.3)
when T > 0, they found that completeextraction of fluid always
occurs with extraction time tE = S(0)/Q such that a(tE) = 0, b(tE)
< 1.This naturally led them to consider the limiting case T → 0
where, combining the previous twoobservations, cusps form in ∂Ω at
time t∗ < tE , and persist until time tE . This corresponds toa
degenerate case of equation (4.3), where K(b) on the right-hand
side is singular as b ↑ 1, butT → 0 to counteract this effect. The
net result is that b(t) is “pinned” at 1 for t > t∗, while
from(4.2), a(t) evolves according to
d
dt[πa2(1 + 1/n)] = −Q,
until t = tE ; this analysis also shows that ZST theory holds
for 0 < t < t∗.These solutions are not classical, because the
free boundary is nonanalytic. There is an embry-
onic theory of such (so-called) “weak solutions” which is
outlined in Appendix A (and which issimilar in spirit to a theory
advanced by Hohlov et al. [15] for the T → 0+ limit of the
Hele-Shawproblem with suction). In generality it demands too much
technical machinery to describe here(in particular, its motivation
requires more of the complex variable theory than we give) and
werefer to [7] for the details. Our aim in this paper is to
illustrate the dynamics of motion on theunivalency boundary, and
the sort of things that can happen. Indeed, given the tentative
natureof the general “weak solution” theory, the direct approach of
taking the limit T → 0 which weadopt provides a useful verification
of the theory, which is shown to be consistent in Appendix A.
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6 L.J. Cummings and S.D. Howison
The weak solutions in the example of HR‘95 are all of similarity
type for t∗ < t < tE , since onlythe scaling parameter a
changes; the shape of the free boundary remains the same
throughout.So, for instance, the solution for n = 2 evolves as a
shrinking cardioid for times t > t∗ in the limitT → 0.
The precise nature of the singularity in (4.3) as b ↑ 1 is
determined by the asymptotic behaviour
K(1− ²) ∼ −12
log(²/8) as ²→ 0,
(see for example [3]). This means that for small positive
surface tension T , the quantity (1 − b)must be exponentially small
in 1/T before surface tension effects become important, a fact
borneout in experiments [19], where ‘almost-cusps’, having radii of
curvature which are exponentiallysmall in the capillary number, can
be observed.
5 The cubic polynomial map
5.1 Real coefficients
We now consider the ideas above applied to the more complicated
case of a general cubic poly-nomial mapping function in the limit T
→ 0. The ZST case of the analogous Hele-Shaw problemwas solved by
Huntingford in [18]. As explained, we expect the evolution of the
free boundary∂Ω(t) to follow ZST theory until the “blow-up time”
t∗, at which point we relax the restrictionon ∂Ω(t) to permit
solutions with persistent cusps in the free boundary. The mapping
functionwe consider is
f(ζ, t) = a(t)(ζ +
b(t)2ζ2 +
c(t)3ζ3
); (5.1)
without loss of generality we assume the scaling factor a to be
real and positive for all time.By suitably rotating the
co-ordinates in the initial domain Ω(0), the general case with both
b(0)and c(0) complex may be reduced to an initial map f(ζ, 0) with
just one complex coefficient. Forsimplicity we shall assume b(0),
c(0) ∈ R, which will then ensure b(t) and c(t) are also real fort
> 0; this is equivalent to the assumption that Ω(t) is symmetric
about the x-axis. We return tothe limitations of our assumption in
§5.2. For this case, (3.2), (3.3) and (3.5) yield the
evolutionequations
d
dt
[a2
(1 +
b2
2+c2
3
)]= −Q
π, (5.2)
d
dt
[a2b
(1 +
2c3
)]= − T
2µG+(0, t)a2b
(1 +
23c
)− T
3µG′+(0, t)a
2c , (5.3)
d
dt[a2c] = −T
µG+(0, t)a2c . (5.4)
5.1.1 The ZST case
With T = 0 these equations are valid until the time t∗ at which
the map ceases to be univalent. Asin [18] we must consider the
domain V in (b, c)-space for which (5.1) is univalent4 on |ζ| ≤ 1,
andfind the phase trajectories of the system (5.3), (5.4) within V
. Cowling & Royster [4] determinedthis univalency domain, in
the more general case of complex coefficients. For real
coefficients, Vis symmetric about the c-axis (so we lose nothing by
restricting attention to the right-half plane
4 Note, though, that we are actually considering the projection
of a “univalency cylinder” in (a, b, c)space, onto a = 1, with
(5.2) providing the extra information about the variation of a with
time.
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Two-dimensional Stokes flow with suction and small surface
tension 7
0.25 0.5 0.75 1 1.25 1.5 1.75
-1
-0.5
0.5
1
c
b
V
Figure 1. The univalency domain V in (b, c)-space for the cubic
polynomial mapping function.
b > 0), and is bounded in b > 0 by the lines
c = 1, b = 1 + c, andb2
4+ 4
(c
3− 1
2
)2= 1. (5.5)
The domain V is shown in Figure 1. Figure 2 shows free boundary
“blow-up” shapes for variousparameter values on ∂V . The line b =
1+c corresponds to formation of a single 3/2-power cusp on∂Ω(t),
except for the isolated points (0,−1) (where we have two 3/2-power
cusps, symmetricallyplaced about both axes), and (8/5, 3/5) (where
we have a single 5/2-power cusp). The line c = 1corresponds to two
3/2-power cusps on ∂Ω (symmetrically placed about both axes when b
= 0),and the elliptical segment of ∂V (which extends from b = 8/5
to b = 4
√2/3) corresponds to loss
of univalency in which the free boundary begins to overlap
itself.Equations (5.3) and (5.4) give the ZST phase paths within V
as the curves
b
c
(1 +
2c3
)= constant = k, (5.6)
for various k ∈ R. In contrast to the Hele-Shaw result of [18],
we find no phase paths whichmeet ∂V tangentially and then re-enter
V ; all ZST solutions blow up with the phase path hitting∂V
obliquely (see Figure 4). A tangent phase path would be associated
with the instantaneousformation of a cusp of (4N + 1)/2-power type,
which would immediately smooth (the phase pathre-entering V ) and
the free boundary would become analytic again. Examples of such
behaviourare known for Stokes flow [24] and the Hele-Shaw problem
[15,16,18] involving transient 5/2-powercusps, but it does not
occur here.
The ZST evolution is then fully determined, the general picture
being that the free boundaryof the fluid domain evolves through a
series of smooth free boundary shapes towards a cusped
orself-overlapping shape (depending on the initial shape ∂Ω(0)) of
the kind illustrated in Figure 2.We emphasise that in no instance
can all the fluid be extracted from Ω(t) in the ZST case. The
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8 L.J. Cummings and S.D. Howison
FluidFluid
FluidFluid Fluid
Fluid
FluidFluid Fluid
(3a)(2)(1)
(3b) (4a) (4b)
(7)(6)(5)
Air Air
Figure 2. Free boundary shapes described by the map (5.1) for
various points (b, c) on the boundary∂V of the univalency domain.
The values used are: (b1, c1) = (0, 1), (b2, c2) = (1, 1), (b3, c3)
= (4
√2/3, 1),
(b4, c4) = (1.8, 0.8461), (b5, c5) = (8/5, 3/5), (b6, c6) = (1,
0), and (b7, c7) = (1/5,−4/5). Pictures (3b)and (4b) are
magnifications of the nonunivalent region, showing how the free
boundary begins to overlapitself; the former case is cusped and
self-overlapping, while the latter is smooth. The value a = 1 was
usedto generate each picture, hence the shapes do not have equal
areas.
phase-paths within (b, c)-space for this ZST solution are
included in Figure 4, which clearly showswhere the blow-up
occurs.
5.1.2 The case T > 0 and its limit as T → 0+
We now consider the effect of small positive surface tension, as
we approach ∂V along a phasepath. Using the definition (2.5), we
are able to find exact expressions for G+(0, t) and G′+(0, t)in
terms of elliptic integrals. These exact expressions are necessary
if we wish to consider theproblem with O(1) surface tension (and
may be used in the numerical solution of the O(1) surface
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Two-dimensional Stokes flow with suction and small surface
tension 9
tension problem) but are not very illuminating for the present
discussion of the limit T → 0; hencewe relegate the details to
Appendix B. The main point to note is that they are singular only
onthe straight-line portions of ∂V (i.e. those portions
corresponding to blow-up via cusp formationrather than by
overlapping) and so only in the neighbourhood of these lines are
surface tensioneffects significant, justifying our earlier
assumption that ZST theory is adequate for t < t∗. To findthe
phase paths near ∂V , we combine (5.3) and (5.4), writing a2c = A,
and a2b(1 + 2c/3) = B,to give
dB
dA=
B
2A+
G′+(0, t)3G+(0, t)
, (5.7)
and hence it is only the ratio of G′+(0, t) to G+(0, t) that is
important.We need to consider two separate cases, according as to
whether the ZST solution breaks down
by reaching the boundary of the univalency domain at c = 1, or
at b = 1 + c (refer forward toFigure 4). Consider first the class
of solutions for which c ↑ 1 within ∂V , along a ZST phasepath. The
crucial point to note is that once c has reached a value close to
1, it is ‘trapped’ nearc = 1 until either the solution blows up
(with c = 1 and attendant cusp formation, or with c ' 1,b ' 4√2/3
on the elliptical portion of ∂V , and self-overlapping of the free
boundary), or untilall fluid is extracted. The reason for this is
simply that if c decreased much below 1, ZST theorywould again take
over, forcing it back up towards c = 1 on a ZST phase path. It
follows that onlya and b vary appreciably with time, and so A ≈ a2,
B ≈ 5a2b/3. Essentially what we are doing isreplacing one of the
equations (5.2)–(5.4) with the requirement that the T = 0+
phase-path mustfollow the univalency boundary ∂V once it has
reached it.
The results of Appendix B show that near c = 1,
G′+(0, t)G+(0, t)
≈ −b ≈ −3B5A
,
hence (5.7) becomes
dB
dA≈ 3B
10A,
giving B = (const.)×A3/10, or, in terms of the mapping function
parameters,b ≈ (const.)× a−7/5, c ≈ 1. (5.8)
Knowing that ZST theory will hold until c ≈ 1, we may take t∗
(the ZST “blow up” time) tobe zero without loss of generality and
proceed from there, so that, in the limit T → 0, c(t) ≡ 1throughout
the motion. Thus, from (5.2) and (5.8), the equations to be solved
are
a2(
43
+b2
2
)=S(0)−Qt
π= a2∗
(43
+b2∗2
)− Qt
π, (5.9)
andb = b∗
(a∗a
)7/5, (5.10)
where we use S(t) to denote the area of Ω(t), and a∗, b∗ denote
the starting values of a andb (0 ≤ b∗ < 4
√2/3 and c∗ = 1, remember). The right-hand side of (5.9) is
simply a linearly
decreasing function of time, reaching zero at the “extraction
time” tE = S(0)/Q. Substitutingfrom (5.10) in (5.9) gives
G(b)− G(b∗) = − 6Qtπa2∗b
10/7∗
, where G(b) := b−10/7(8 + 3b2). (5.11)
Now, G(b) is positive and monotone decreasing in b on the range
of interest (namely 0 ≤ b ≤4√
2/3), so (5.11) tells us that b must be monotone increasing in
t, from its starting value b∗.Hence the phase path must follow the
line c = 1 in this direction, ending either at time tE , or
-
10 L.J. Cummings and S.D. Howison
when it reaches b = 4√
2/3. Complete extraction cannot occur in this regime, since
(5.9) and(5.10) give the area of the fluid domain as
S(t) = π
(43a2 +
b2∗a14/5∗
2a4/5
),
which is always positive. Hence we deduce that the phase path
reaches b = 4√
2/3 before all thefluid has been extracted, and the solution
breaks down with ∂Ω(t) beginning to overlap itself(Figure 2, (3a)
and (3b)).
We now consider the case of solutions approaching the
straight-line portion b = 1 + c of ∂Valong a ZST phase path,
observing, by the same argument as above, that a phase path is
‘trapped’near this line once it is sufficiently close to it (Figure
4). We may thus eliminate either b or cin the ZST limit, and we
choose to work with b (so c = b − 1). In this case, A ≈ a2(b − 1)
andB ≈ a2b(2b+ 1)/3. The asymptotic evaluation of the ratio G′+(0,
t)/G+(0, t) as the line b = 1 + cis approached is performed in
Appendix B. This limit is found to be nonuniform on the rangeb ∈
(0, 8/5), being equal to −2 everywhere except at the single point b
= 0. Thus for b > 0 (5.7)becomes
dB
dA≈ B
2A− 2
3,
which has solution
B = −4A3
+ λ√|A|,
for some constant λ. We again take t∗ = 0 without loss of
generality, and our initial conditionsmust satisfy b∗ = 1 + c∗
(where now 0 ≤ b∗ ≤ 8/5). In terms of a and b then, we have
ag(b)|b− 1|1/2 =
a∗g(b∗)|b∗ − 1|1/2
≡ 3λ,
where g(b) := 2b2 +5b−4, holding together with the mass
conservation equation (5.2). After somerearrangement, and putting c
= b− 1 (since we remain on this part of the univalency
boundary),(5.2) becomes
(a
a∗
)2h(b)− h(b∗) = −6Qt
πa2∗,
for h(b) defined by
h(b) := 5b2 − 4b+ 8 ≡ 6(
1 +b2
2+
(b− 1)23
).
Combining the previous two equations, eliminating the ratio a/a∗
between them, we finally arriveat an analogue of (5.11),
|b− 1| h(b)g(b)2
− |b∗ − 1| h(b∗)g(b∗)2
= −6Qtπa2∗
|b∗ − 1|g(b∗)2
. (5.12)
Ignoring the two exceptional cases b∗ = 1, λ = 0 for the moment
(on our range of interest, λ = 0occurs if and only if b∗ = bc = (−5
+
√57)/4), we see that the right-hand side of (5.12) is a
monotone decreasing function of time, and so too must be the
left-hand side, so that F (b) :=|b− 1|h(b)/g(b)2 decreases with
time. The area of the fluid domain is given by
S(t) =πa2∗h(b∗)
6F (b)F (b∗)
,
so complete extraction occurs if and only if F (b) falls to
zero; this corresponds to extraction timetE = πa2∗h(b∗)/(6Q). A
plot of F (b) on (0, 8/5) is given in Figure 3 (b = 8/5 is the
point at which
-
Two-dimensional Stokes flow with suction and small surface
tension 11
0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
1.1 1.2 1.3 1.4 1.5 1.60
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Figure 3. The function F (b) governing evolution on the part b =
1 + c of ∂V . The singularity occurs atb = bc = (−5 +
√57)/4. (Note the difference in scales between the two
plots.
the form of ∂V changes, the small elliptical portion of ∂V for
8/5 < b < 4√
2/3 corresponding toblow up of solutions by overlapping of the
free boundary). Important features to note are that:
• F (b) vanishes only at b = 1;• F (b) has a singularity at b =
bc, corresponding to a critical point in the phase diagram;• F (b)
is monotone increasing (to infinity) on (0, bc), monotone
decreasing (to zero) on (bc, 1),
and monotone increasing on (1, 8/5);• F ′(b) = 0 at b = 8/5, and
only there, corresponding to the formation of the 5/2-power cusp.We
must have F (b) decreasing with t, hence for b∗ ∈ (bc, 1) and b∗ ∈
(1, 8/5), the phase path willapproach the point b = 1, c = 0, with
complete extraction occurring as we reach this point. Bycontrast,
if b∗ ∈ (0, bc) we must have the phase path approaching b = 0, c =
−1. Since F (0) > 0,this point is reached before all the fluid
has been extracted, but due to the symmetry of the phasediagram
about the c-axis, we are forced to stay at this point. (For the
moment we ignore thecomplications hinted at by the nonuniformity of
the limit G′+(0, t)/G+(0, t) at this point.)
Recall now the comment in footnote (4), that we have actually
been considering the projectionof a univalency cylinder by
suppressing the parameter a. We are thus in one of the special
casesconsidered in HR’95; the subsequent evolution is of the
‘similarity’ type discussed there, with b ≡ 0,c ≡ −1, and the
parameter a changing in accordance with the corresponding mass
conservationequation. The full phase diagram in the (b, c)-plane is
given in Figure 4, with phase paths thatare in some way ‘special’
represented by dashed lines. The bold arrows indicate the sense in
whichthe phase paths ‘turn around’ as they hit ∂V .
It is now apparent that the ‘exceptional cases’ b∗ = 1, b∗ = bc
mentioned earlier are stableand unstable (respectively) critical
points in the phase diagram, and thus also represent
possible‘similarity’ solutions of the kind studied in HR’95, the
dotted phase path c = 0 being exactly oneof those solutions. Note
that for this special solution, reaching b = 1 no longer need be
synonymouswith total extraction, since the right-hand side of
(5.12) is now identically zero; indeed, by theanalysis of HR’95 we
do remain a finite time at (1, 0) before extraction is complete.
The points(0,−1) and (0, 1) are also critical points, stable (but
see §5.2) and unstable respectively, andagain, are members of the
family of similarity solutions of HR’95. We may summarise our
resultsas follows:
• Phase paths which hit ∂V at (1, 0) or (0,−1) terminate there
and represent stable similaritysolutions, since adjacent phase
paths are also entering these points.
• Paths which hit ∂V at (0, 1) and (bc, 1− bc) terminate there
and represent similarity solutionswhich are unstable, since
neighbouring paths are diverging.
-
12 L.J. Cummings and S.D. Howison
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
c
b
Blow-upby overlapping
only on this curvedportion of ∂V
5/2-power cusp
Complete extraction
Critical point bc
forms here
occurs here
Two 3/2-power cusps in ∂Ω along line c = 1
One 3/2-power cusp in ∂Ω along line b = 1 + c (b 6= 0, 1,
8/5).Two 3/2-power cusps (symmetrically placed) at b = 0, c =
−1.
Figure 4. The univalency diagram (restricted to the right-half
(b, c)-plane) for the T → 0 solution withthe cubic polynomial
mapping function (5.1). The shaded region (outside V ) corresponds
to a nonunivalentmap. The phase paths do not stop when they reach
the boundary ∂V , but turn around and follow ∂V ;this is indicated
by the bold arrows.
• Paths for which c∗ = 1, b∗ ∈ (0, 4√
2/3) turn to the right and follow ∂V along c = 1,
reaching(4√
2/3, 1) before extraction is complete, at which point the free
boundary begins to overlapitself. The present analysis then no
longer applies, and the solution cannot be continued.
• Paths hitting the curved portion of ∂V likewise represent
self-overlapping fluid domains, andcannot be continued.
• Paths for which c∗ = b∗ − 1, 1 < b∗ < 8/5 turn around
and enter the point (1, 0); reaching thispoint is simultaneous with
complete extraction.
• Ditto for bc < b∗ < 1.• Paths for which c∗ = b∗ − 1, 0
< b∗ < bc turn around and enter (0,−1), reaching this
point
before extraction is complete; subsequent evolution is of
‘similarity’ type as discussed in HR’95.
The 5/2-power cusp is an interesting borderline case, being the
point of transition between ZSTsolutions which break down via
formation of a 3/2-power cusp, and those which break down
viaoverlapping of the free boundary. This path must still turn
around and enter (1, 0); however, thefact that F ′(8/5) = 0 (Figure
3) implies that this path ‘only just makes it’. Geometrically,
the5/2-power cusp immediately becomes a 3/2-power cusp, which then
persists.
-
Two-dimensional Stokes flow with suction and small surface
tension 13
The existence of the point bc is also interesting. As we move
along ∂V from (1, 0) towards(0,−1), ∂Ω(t) evolves continuously from
a cardioid (with a cusp on the left-hand side), to a fullysymmetric
shape having cusps on both sides. As is does so, a ‘dimple’
develops on the right-handside (see (6) and (7) in Figure 2), which
becomes more pronounced, eventually turning into thesecond cusp at
the point (0,−1) on ∂V . It is perhaps not surprising then that
there is some criticalpoint beyond which the ‘dimple’ is too large
to disappear, and the ultimate shape has to havetwo cusps. If the
dimple is small enough (i.e. b∗ > bc), then the ultimate shape
will have just onecusp. For the solutions with c∗ = 1, however, the
possible geometries are such that the two-cuspstate is always
unstable, and ultimate overlapping of the free boundary has to
occur.
5.2 Complex coefficients
Recall our earlier comment, that the assumption of real
coefficients in the mapping function (5.1)was equivalent to
assuming symmetry of Ω(t) about the x-axis. The results obtained
seem to havea remarkably rich structure nonetheless; however they
are somewhat deceptive, as considerationof the case with complex
coefficients reveals.
A little thought about the conclusions of §5 reveals an apparent
contradiction: the point (0,−1)in (b, c)-space is stated to be a
stable equilibrium point, whilst the point (0, 1) is an
unstableequilibrium point. But the two configurations are actually
identical, one being a rotation throughangle π/2 of the other. In
fact, the conclusions regarding the point (0,−1) were a little
suspectanyway, since we knew the limit G′+(0, t)/G+(0, t) to be
nonuniform at this point, but the analysisaway from this point did
indicate that it should be a stable equilibrium.
In the preceding analysis, we have been considering a single,
two-dimensional cross-section ofwhat is actually a four-dimensional
univalency domain V4 in complex (b, c)-space.5 Determinationof this
domain is the subject of Cowling & Royster’s (henceforth C
& R) paper [4]. There, theauthors note that the cross-section
={b} = 0 of V4 is symmetric about the planes
-
14 L.J. Cummings and S.D. Howison
a2(b1 +
23(b1c1 + b2c2)
)= k3, (5.18)
a2(b2 +
23(b1c2 − b2c1)
)= k4, (5.19)
which, as the four arbitrary constants k1 to k4 vary, give paths
in (b1, b2, c1, c2)-space (afterelimination of a). We now recall
the statement of C & R that it is sufficient to consider
thesituation b2 = 0, b1 > 0. Suppose we seek such solutions to
the above equations (5.16)–(5.19).The first two are unchanged,
whilst the second two become
a2b1
(1 +
2c13
)= k3, (5.20)
a2b1c2 = k4. (5.21)
Equations (5.16) and (5.17) givec1c2
= constant,
whilst (5.20) and (5.21) give
1c2
+2c13c2
= constant,
which together imply that either both c1 and c2 must be
constant, or else c2 = 0. Supposing thefirst case, with c2 6= 0,
then to satisfy the equations we need both a and b1 to be constant
also, inwhich case the mass conservation equation cannot hold
(except in the trivial case Q = 0). Hencewe must have c2 = 0,
showing that the only family of solutions for which b ∈ R
throughout theevolution, is that already found for which c ∈ R
also.
The result of C & R essentially says that restricting
attention to V3 yields all possible freeboundary shapes, the
remainder of V4 consisting of rotations and reflections of shapes
which arecontained within V3. For a map with constant coefficients
it is then sufficient to consider V3, sinceany free boundary
configuration can be generated by some point within V3 provided the
axesare suitably chosen. With time-dependent coefficients, we may
choose axes such that the initialconfiguration Ω(0) is generated by
a point of V3; however the above shows that only if c2 = 0 isthe
configuration for t > 0 also generated by a point of V3.
Solution trajectories for c2 6= 0 migrateto regions of V4 outside
V3.
C & R’s observation is therefore of limited use, since the
only family of solution trajectorieslying wholly within the
three-dimensional cross-section V3 ⊂ V4 is the family of real
solutionsalready studied—all other solution trajectories simply
intersect V3 at a single point. The full four-dimensional space V4
is horribly difficult (if not impossible) to determine and study
analytically.We consider instead whether we might find a
three-dimensional solution family for the case inwhich c is real,
but b is complex. Setting c2 = 0 in equations (5.16)–(5.19)
gives6
a2c1 = k1, k2 = 0,
a2b1
(1 +
2c13
)= k3, a2b2
(1− 2c1
3
)= k4;
since (5.17) has reduced to an identity, we are able to
eliminate a from these equations to find
6 Note that this analysis also covers the case c1 = 0, c2 6= 0,
since if we make the substitutionsB1 = (b1 + b2)/
√2, B2 = (b2 − b1)/
√2, ξ = exp(−iπ/4)ζ and F = exp(iπ/4)f we have
F (ξ, t) = a
ţξ +
1
2(B1 + iB2)ξ
2 +c23
ξ3ű
,
so we get the same phase diagram in (B1, B2, c2)-space.
-
Two-dimensional Stokes flow with suction and small surface
tension 15
phase trajectories in (b1, b2, c1)-space: these are determined
by the two equations
b1c1
(1 +
2c13
)= constant,
b2c1
(1− 2c1
3
)= constant. (5.22)
To get an idea of this three-dimensional cross-section V∗ of V4,
we consider simple two-dimensionalcross-sections. The cross-section
b2 = 0 is the case already studied (the domain V given by
(5.5)).The cross-section b1 = 0 corresponds to maps of the form
f(ζ, t)a
= ζ +ib22ζ2 +
c13ζ3.
Making the substitutions ζ = −iζ̂, c1 = −ĉ1 and f(ζ, t) =
−if̂(ζ̂, t) we find thatf̂(ζ̂, t)a
= ζ̂ +b22ζ̂2 +
ĉ13ζ̂3,
so the intersection of V∗ with this cross-section is exactly the
domain V , but inverted with respectto c1; call it V†. Likewise, we
have a ZST solution family lying entirely within V†, with phase
pathsexactly as for the real coefficients case, but inverted with
respect to c1. The T → 0 limit is alsoinferred from the earlier
analysis.
The other two-dimensional cross-section of V∗ we can look at is
c1 = 0. This is particularlyeasy, the map now being
f(ζ, t)a
= ζ +(b1 + ib2)
2ζ2,
so that f ′(ζ, t) = 0 only if ζ = −1/(b1 + ib2), and the map is
univalent on the discb21 + b
22 ≤ 1.
A solution family again lies in this cross-section (which we
call V‡), with solution trajectorieswhich are straight lines
b1b2
= constant,
as can be seen from (5.18) and (5.19) with c1 = 0 = c2. All
points on the univalency boundary areequivalent, in the sense that
the free boundary shapes represented by the maps are just
rotationsof the same cardioid. The T → 0 limit of this solution
family is of the “similarity solution” type,with initial limaçons
becoming cardioids (before all the fluid has been extracted) which
thenpersist in a self-similar fashion until extraction is
complete.
The schematic Figure 5 indicates how the three-dimensional
domain V∗ fits together. Giventhe equivalence of the cross-sections
V and V†, we now see plainly the equivalence of the points{b = 0, c
= 1} and {b = 0, c = −1}, and the arrows in Figure 5 show how this
configurationdestabilises.
The only point in our phase diagram (Figure 4) which is possibly
still ambiguous is b = 1, c = 0,which is claimed to be stable. How
do we know that this point does not ‘destabilise’ like the pointb =
0, c = 1, by the phase paths moving onto ∂V4 6⊂ ∂V ? Consider again
the three-dimensionalsubset V∗ ⊂ V4, which we have seen contains a
solution family. We know that phase paths of thisfamily starting
within V∗ move out to the boundary under the ZST evolution
equations (5.22),and then move over the surface ∂V∗ under the
influence of the T = 0+ effect (cf. the pathsmoving along the
boundary ∂V in Figure 4). From our experience with the real
coefficients case,we know that a phase path can leave ∂V∗ through a
critical point, and move onto other partsof ∂V4. However, the point
is that V∗ contains a solution family, and therefore the phase
pathswithin V∗ (and on ∂V∗) must be topologically consistent, just
as they are within V . It is easilychecked (by sketching the phase
paths on the surface of some suitable three-dimensional object—a
plastic cup, in the authors’ case!) that the only consistent
possibility is for all phase paths to
-
16 L.J. Cummings and S.D. Howison
-1
0
1
-0.5
-0.25
0
0.25
0.5-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-0.5-0.25
0
0.25
0.5
-1
0
1
-1
-0.5
0
0.5
1
-0.5-0.25
0
0.25
0.5
-1
0
1
-1
-0.5
0
0.5
1-1
-0.5
0
0.5
1
-0.5
-0.25
0
0.25
0.5
-1
-0.5
0
0.5
1
-1
0
1
-1
0
1
-1
-0.5
0
0.5
1
-1
0
1
V†
b1
b2
V
V‡V∗
c1
Figure 5. The three-dimensional univalency domain V∗ ⊂ V4, and
its two-dimensional cross-sections V ,V† and V‡. The arrows on V†
indicate how the point {b = 0, c = −1} destabilises (cf. Figure
4).
-
Two-dimensional Stokes flow with suction and small surface
tension 17
enter the ring {b21 + b22 = 1, c1 = 0} on ∂V∗ (all points of
which represent equivalent free boundaryshapes). Thus, all such
solution shapes are stable with respect to perturbations in c1.
Also, asobserved in footnote 6, there is an analogous family of
solutions with c1 ≡ 0 and c2 6= 0, which iscontained within the
same domain V∗ in (B1, B2, c2)-space. In the same way then,
solution shapeson {B21 +B22 ≡ b21 + b22 = 1, c2 = 0} are stable to
perturbations in c2. Thus, the “cardioid” shapeswith c ≡ 0 are the
attractors of the cubic polynomial family we consider, and in this
sense thestability of the point b = 1, c = 0 in Figure 4 is
confirmed.
Our study is clearly not exhaustive, since we have considered
only a limited subset V∗ of V4,which contains a family of solutions
of which the real coefficients case is a sub-family. In fact
thissub-family appears twice within V∗, as we have seen (in V and
in V†) so there is considerablerepetition even within this limited
subset. V∗ does not contain all possible free boundary
con-figurations. On the other hand, the cross-section V3 (b2 = 0)
of V4 is a minimal set of possiblefree boundary shapes if axes are
chosen appropriately, but complex-parameter solutions do not
liewholly within this space. Evolution in time cannot be determined
by studying V3 then, unless theanalysis is somehow modified to
allow the co-ordinates within the fluid domain to rotate suitablyin
time—we do not consider this possibility.
6 Conclusion
We have discussed in detail the evolution of the weak solution
for a cubic map ZST Stokes flowsolution. We have confirmed that the
weak solution is the usual ZST solution before blow-up, andwe have
analysed its progress around the boundary of the univalency domain
at later times. Forthis particular map, the free boundary can have
either one or two cusps when it is singular, andwe have presented
strong evidence (short of a full examination of the
four-dimensional univalencydomain, which does not appear feasible)
that if all the fluid is extracted, the ultimate shape
isgenerically a cardioid with one cusp, the two-cusped
configurations being unstable or leading toself-overlapping. It is
an interesting open question how many terminal cusps might be
generic fora more complex map.
AcknowledgementsThe authors are grateful for helpful
conversations with Prof. J.R. King and Dr J.R. Ockendon.
Appendix A The “weak solution” theory
In this Appendix we briefly outline the “weak solution” concept
advanced in [7], and its applicationto the cubic-mapping problem.
Roughly speaking, from (2.3) and (2.5) one can see that
naivelysetting T = 0 in the govering equations gives the correct
ZST limit as long as the free boundaryis smooth (f ′(ζ) 6= 0 on |ζ|
= 1, so t < t∗). However as t → t∗ and a zero (or zeros) of f
′(ζ, t)approaches the unit circle (at ζ = ζ∗, say) the function
G+(ζ, t) becomes infinite as (ζ, t) →(ζ∗, t∗). Thus we get “zero
times infinity” in equation (2.3): T → 0 is a singular limit. In
[7] it isargued that if, in the ZST problem, cusps form at points
(ζ∗j , t
∗j ) in (ζ, t)-space, then in the limit
T → 0 we should make the substitutionT
2µG+(ζ, t) 7→
∑
j
Gj(t)
(ζ∗j + ζζ∗j − ζ
), (A 1)
where the functions Gj(t) must be determined (by the usual
singularity-matching) but are realand satisfy
Gj(t) ≡ 0 for t < t∗j .
-
18 L.J. Cummings and S.D. Howison
Thus the Gj(t) act as “switches” for the singularities at ζ∗j
which appear at time t∗j . Looking
back at the definition of G+(ζ, t) in (2.5), one can see that (A
1) is a reasonable ansatz, since asthe singularity formation times
t∗j are approached the major contributions to the integral comefrom
the points ζ = ζ∗j . Matching the singular behaviour in equation
(2.3) at the new singularitiesimmediately gives the result f ′(ζ∗j
, t) = 0 for t > t
∗j , that is, the cusps persist. In this general
theory, more than one “breakdown time” t∗ is permitted because
in the weak formulation thesolution does not break down at time t∗
but continues, with non-analytic free boundary. Notethat the ζ∗j
may themselves be functions of t, i.e. the cusps may move along the
free boundary(and do, in our example).
Thus in the T → 0 limit the governing equation (2.3) is replaced
by its equivalent with thesubstitution (A 1), and (as mentioned in
§2) solutions to the model may be obtained by matchingthe singular
behaviour on both sides of the new equation (for our example the
only singularityother than those at the cusps is at ζ = 0). However
we may circumvent this singularity-matchinganalysis by observing
that in line with the substitution (A 1) we must have
T
2µG′+(ζ, t) 7→ 2
∑
j
Gj(t)ζ∗j(ζ∗j − ζ)2
.
Thus instead of evaluating the limit G′+(ζ, t)/G+(0, t) as the
univalency boundary is neared (whichwas the direct approach in §5;
see equation (5.7)), we need to determine
L :=2
∑(Gj(t)/ζ∗j )∑Gj(t)
, (A 2)
for t > t∗ in the two breakdown cases.
Case (I): Breakdown by approaching the line c = 1
In this case two cusps form simultaneously. In the notation
introduced above we have (by sym-metry)
ζ∗1 = ζ∗, ζ∗2 = ζ̄∗, t
∗1 = t
∗2 = t
∗, G1(t) = G2(t) = G(t),
where ζ∗ = (−b+ i√b2 − 4)/2. Thus since |ζ∗| = 1 the definition
(A 2) givesL = 2
-
Two-dimensional Stokes flow with suction and small surface
tension 19
and G′+(0)≡2πa
∫ π0
cos θ dθ
[b2 + (c− 1)2 + 2b(1 + c) cos θ + 4c cos2 θ]1/2,
the forms of which functions change as we cross the curve b2 =
4c in V (according to whether thedenominator has real or complex
roots as a function of cos θ); G+(0) itself is continuous
acrossthis curve, however. In b2 < 4c we use formula 3.145.2 in
Gradshteyn & Ryzhik [10] (henceforthG & R), and also the
asymptotic result
K(1− ²) ∼ −12
log(²/8) ∼ −12
log ² as ²→ 0, (B 1)
where K( · ) is the complete elliptic integral of the first
kind. We find that
G+(0) =2K(k1)
πa√
(c+ 1)2 − b2 , where k21 :=
4c− b2(c+ 1)2 − b2 ,
∼ −2πa√
4− b2 log(1− c) as c ↑ 1,
c = 1 being the only singularity within this part of V . In b2
> 4c we need formulae 3.147.6 and3.147.4 of G & R (in
regions c > 0, c < 0 respectively) together with (B 1) to
deduce that
G+(0) =2K(k2)
πa(√b2 − 4c+ (1− c)) where k
22 :=
4(1− c)√b2 − 4c[√b2 − 4c+ (1− c)]2
∼ −1πa(1− c) log(1 + c− b) as (1 + c− b) ↓ 0,
b = 1 + c now being the only line of singularities within V .
Explicit formulae for G′+(0) are muchmore complicated; in b2 <
4c we find
G′+(0) =−b(1 + c)K(k1)πac
√(1 + c)2 − b2 +
2√c{E(k1)F (ψ, k′1) +K(k1)(E(ψ, k′1)− F (ψ, k′1))} ,
where (k′1)2 = 1− k21, ψ = sin−1(
b
2√c); (B 2)
here k1 is as previously defined, E( · ), E( · , · ) denote the
complete and incomplete (respectively)elliptic integrals of the
second kind, and F ( · , · ) denotes the incomplete elliptic
integral of thefirst kind (so K( · ) ≡ F (π/2, · )). The key
formulae used in finding this expression were 259.07and 410.02 in
Byrd & Friedman [3] (henceforth B & F), along with various
properties of ellipticintegrals and Jacobian elliptic functions,
all of which may be found in B & F.
In b2 > 4c we find
G′+(0) =4
πa(√b2 − 4c+ (1− c))
{AK(k2)− (1 +A)Π
(2
1−A, k2)}
,
where A =14c
(−b(1 + c) + (1− c)
√b2 − 4c
); (B 3)
again, k2 is as previously defined, and Π( · , · ) denotes the
complete elliptic integral of the thirdkind. In finding this
expression the formulae used were G & R 3.148.6 and 3.148.4 (in
regionsc > 0 and c < 0 respectively).
In using these two books, care was necessary to account for
slight differences in definitions.Likewise, when carrying out
numerical checks on the analysis, care was needed due to
differentinbuilt definitions in the software package Mathematica.
The above assumes the definitions:
F (φ, k) =∫ φ
0
dθ√1− k2 sin2 θ
=∫ sin φ
0
dx√(1− x2)(1− k2x2) , (B 4)
E(φ, k) =∫ φ
0
√1− k2 sin2 θ dθ =
∫ sin φ0
√1− k2x2√1− x2 dx ,
-
20 L.J. Cummings and S.D. Howison
Π(α2, k) =∫ π/2
0
dθ
(1− α2 sin2 θ)√
1− k2 sin2 θ
=∫ 1
0
dx
(1− α2x2)√
(1− x2)(1− k2x2) .
The results of §5 require the asymptotic evaluation of the ratio
G′+(0)/G+(0) near each of thelines c = 1 and b = 1 + c. This is not
too bad for the case c ↑ 1, and fairly nasty for the caseb ↓ (1 +
c); we give only brief details.
In b2 < 4c results of B & F §§111–112 are used, together
with (B 1) above, to deduce that asc ↑ 1, the term in curly
brackets in G′+(0) (B 2) is everywhere negligible compared to the
firstterm. Hence we see that the asymptotic behaviour here is
G′+(0)G+(0)
∼ −b. (B 5)
To study the behaviour of G′+(0) as b ↑ (1 + c) in the region b2
> 4c we must consider thecases c > 0 and c < 0 separately,
since these give different types of behaviour in (B 3). We write² =
1+c−b and eliminate b to work with c and ², so that letting ²→ 0
corresponds to approachingthe univalency boundary ∂V . We also
define the auxiliary parameter δ := ²2/(4(1− c)2); this willalways
be small since we do not consider the elliptical part of ∂V
corresponding to blow-up viaoverlapping of the free boundary, so c
lies in the range −1 < c < 3/5. We find:
A = −1− 2δ + · · · , −(1 +A) = 2δ + · · · ,
k22 = 1− δ(
1 + c1− c
)2+ · · · , α2 ≡ 2
1−A = 1− δ + · · · .
We know the asymptotic behaviour of the first term in curly
brackets in G′+(0) (B 3), from (B 1).The term outside, multiplying
the curly bracket, is also straightforward. Hence we only need
tofind the behaviour of the second term within curly brackets,
which to lowest order is
−(1 +A)Π(
21−A, k2
)∼ 2δΠ(α2, k2).
Suppose first that c ∈ (0, 3/5). Then by the above expressions,
0 < α2 < k22 < 1, and so accordingto the classifications
of B & F (p. 223) we have a case II elliptic integral of the
third kind (acircular case).7 Formula 412.01 in B & F thus
applies, giving the result in terms of the HeumanLambda function.
Results from §150 of the book may then be used to arrive at the
approximation
2δΠ(α2, k2) =1− c√c
(π
2− sin−1
(1− c1 + c
))+O(δ log δ),
which gives excellent agreement when checked numerically. This
term will thus be everywherenegligible compared to the first term
in the curly brackets (K(k2) being singular as k2 → 1),hence we get
the approximation
G′+(0) ∼2
πa(1− c) log(1 + c− b).
It follows that for c in this parameter range we will have
G′+(0)G+(0)
∼ −2,
as we approach the boundary.For c ∈ (−1, 0) (still using (B 3))
we have 0 < k22 < α2 < 1, which is a case III elliptic
integral7 The case c = 0 is the special case α2 = k2, and provides
a check on the analysis in both regions c > 0,
c < 0.
-
Two-dimensional Stokes flow with suction and small surface
tension 21
of the third kind (a hyperbolic case). Thus formula 414.01 of B
& F applies, and it is relativelyeasy to see that
2δΠ(α2, k2) ' 1− c√−cK(k2)Z(β, k2) for β = sin−1
(α
k2
)=π
2− 2
√−cδ1− c + · · · ,
where Z( · , · ) denotes the Jacobi Zeta function (discussed in
§140 of B & F). Then
G′+(0) ∼−2K(k2)πa(1− c)
(1− 1− c√−c Z(β, k2)
),
so thatG′+(0)2G+(0)
∼ −1 + 1− c√−c Z(β, k2).
Results of §140 and §100 in B & F show that for small δ,
Z(β, k2) ∼ 1K(k2)
log
(2√−c1 + c
+(
1− 4c(1 + c)2
)1/2).
Note that c = 0 is not a problem point, despite the factor 1/√−c
in the above, because for small c
we may expand the logarithmic term appearing in the expression
for Z(β, k2). The only problemis at c = −1; away from this point we
can see that
G′+(0)G+(0)
∼ −2.
Near c = −1, the function Z(β, k2) will no longer be negligible
according to the above. Here wehave
G′+(0)2G+(0)
∼ −1 + 2(
1 +log ²
log(1 + c)
)−1= −1 + 2
(1 +
log ²log(²+ b)
)−1,
for c close to −1 (or, b small and positive). So, for instance,
if we take b = λ² for some order onequantity λ we will have
G′+(0)G+(0)
→ 0 as ²→ 0.
In particular, this will be the case as we approach the
univalency boundary along the phase pathb ≡ 0. We thus have a
nonuniform limit, with
G′+(0)G+(0)
→ −2 as ²→ 0 (B 6)
everywhere except c = −1 (or b = 0); at this point the limit is
zero.
-
22 L.J. Cummings and S.D. Howison
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